Regular partitions of (weak) finite generalized polygons: Part I

We define a regular m-partition of a distance regular graph as a partition of the vertex set into m classes, such that the number of vertices of a given class adjacent to a fixed vertex of another class (but possibly the same), is inde-pendent of the choice of that vertex in this class. Furthermore, we exhibit a technique to determine exact, discrete or bounding values for the intersection numbers of two such regular partitions of a DRG. In the first part of the presentation we define distance regular graphs and generalized polygons and give a well illustrated overview on the substructures of these geometries. As an application to the newly obtained technique, in the second part of the presentation we perform a structural investigation on the substructures of finite generalized polygons and, besides some new results, we give unifying, alternative and more elegant proofs of the results in [1] and [2]. Reference